Hessian potentials with parallel derivatives

TL;DR

This paper classifies functions with non-degenerate Hessians whose derivatives are parallel with respect to the Hessian metric, linking them to Jordan algebras and canonical barriers on symmetric cones.

Contribution

It provides a complete characterization of functions with parallel derivatives in terms of logarithmically homogeneous functions and Jordan algebra structures.

Findings

Solutions are logarithmically homogeneous functions.

Connections established between parallel derivatives and metrised Jordan algebras.

Characterization of canonical barriers on symmetric cones.

Abstract

Let $U \subset \mathbb A^n$ be an open subset of real affine space. We consider functions $F: U \to \mathbb R$ with non-degenerate Hessian such that the first or the third derivative of $F$ is parallel with respect to the Levi-Civita connection defined by the Hessian metric $F"$. In the former case the solutions are given precisely by the logarithmically homogeneous functions, while the latter case is closely linked to metrised Jordan algebras. Both conditions together are related to unital metrised Jordan algebras. Both conditions combined with convexity provide a local characterization of canonical barriers on symmetric cones.